Volume 43 | Number 1 | Year 2017 | Article Id. IJMTT-V43P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V43P507

This paper deals with the asymptotic and oscillatory properties of solutions of a class of second order non-linear damped neutral difference equations of the form D [r(n)(Dz(n))a] + a(n + 1)(Dz(n + 1))a + q(n) f (x(n + s)) = 0; n n0, () where z(n) = x(n)

where is a ratio of positive odd integers, , and
are sequences of real numbers, and are integers, and is a real valued continuous function. We
established some new sufficient conditions under which every solution of is either oscillatory or tends to zero as
. The results are illustrated with examples.

[1] R. P. Agrwal, Difference Equations and Inequalities; Theory, Methods and Applications, Marcel Dekker, New York, 1999.

[2] R. P. Agarwal, P.J.Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht, 1997.

[3] Aleksandra Sternal-Blazej Szmanda, Asymptotic and oscillatory behaviour of certain difference equations, Lematematiche, (1996), 77-– 86.

[4] E. Thandapani, Z. Liu, R. Arul and P. S. Raja, Oscillation and asymptotic behavior of second order difference equations with nonlinear
neutral terms, Applied Mathematics E-Notes, 4(2004), 59-67.

[5] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.

[6] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Reprint of the 1952 edition, Cambridge University Press, Cambridge, UK, 1988.

[7] W. T. Li and S. S. Cheng, Classifications and existence of positive solutions of second order nonlinear neutral difference equations, Funkcial.
Ekvac., 40(1997), 371-393.

[8] W. T. Li, X. L. Fan, Oscillation criteria for second-order nonlinear difference equations with damped term, Comp. Math. Appl., 37(1999)
17–30.

[9] A. Murugesan and K. Ammamuthu, On oscillation solutions of second order neutral difference equations. (Communicated)

[10] M. Peng, Q. Xu, L. Huang, W.G. Ge, Asymptotic and oscillatory behavior of solutions of certain second-order nonlinear difference
equations, Comp. Math. Appl., 37(1999), 9–18.

[11] R. N. Rath, Seshadev Padhi and B. L. S. Barik, Oscillatory and asymptotic behaviour of a homogeneous neutral delay difference
equation of second order, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 3(2008)(3), 453-46.

[12] Z. Szafranski, B. Szmanda, Oscillation theorems for some nonlinear difference equations, Appl. Math. Comp., 83(1997), 43–52.

[13] B. Szmanda, Oscillation theorems for nonlinear second-order difference equations, J. Math. Anal. Appl., 79(1981), 90–95.

[14] E. Thandapani, I. Gyori, B.S. Lalli, An application of discrete inequality to second-order nonlinear oscillation, J. Math. Anal. Appl.,
186(1994), 200–208.

[15] E. Thandapani, B. S. Lalli, Oscillation criteria for a second-order damped difference equation, Appl. Math. Lett., 8(1995), 1–6.

[16] E. Thandapani and K. Mahalingam, Necessary and sufficient conditions for oscillation of second order neutral difference equations,
Tamkang Journal of Mathematics, 34(2)(2003).

[17] E. Thandapani, S. Pandian, Asymptotic and oscillatory behavior of general nonlinear difference equations of second-order, Comp. Math.
Appl., 36(1998), 413–421.

[18] A. K. Tripathy, On the oscillation of second order nonlinear neutral delay difference equations, Electronic Journal of Qualitative Theory of
Differential Equations, 11(2008),
1-12.

[19] E. Tunc and S. R. Grace, On oscillatory and asymptotic behavior of a second order nonlinear damped neutral differential equation,
International Journal of Differential Equations vol 2016, Article ID 3746368.

[20] S. H. Saker and S. S. Cheng, Oscillation criteria for difference equations with damping terms, Appl. Math. Comput., 148(2004), 421-442.

[21] P. J. Y. Wong, R. P. Agarwal, Oscillation and monotone solutions of second-order quasilinear diff erence equations, Funkcial. Ekvac.,
39(1996), 491-517.

A. Murugesan, K. Ammamuthu, "New Oscillation Conditions for Second Order
Non-Linear Neutral Difference Equations with
Damping Term," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 43, no. 1, pp. 33-44, 2017. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V43P507